Heat transfer from convecting-radiating fin through optimized Chebyshev polynomials with interior point algorithm

Abstract In this paper, the problem of determining heat transfer from convecting-radiating fin of triangular and concave parabolic shapes is investigated.We consider one-dimensional, steady conduction in the fin and neglect radiative exchange between adjacent fins and between the fin and its primary surface. A novel intelligent computational approach is developed for searching the solution. In order to achieve this aim, the governing equation is transformed into an equivalent problem whose boundary conditions are such that they are convenient to apply reformed version of Chebyshev polynomials of the first kind. These Chebyshev polynomials based functions construct approximate series solution with unknown weights. The mathematical formulation of optimization problem consists of an unsupervised error which is minimized by tuning weights via interior point method. The trial approximate solution is validated by imposing tolerance constrained into optimization problem. Additionally, heat transfer rate and the fin efficiency are reported.


Preliminaries and problem formulation
The heat dissipation mechanism considered in literature is either pure convection or pure radiation. In applications where ns operate in a free or natural convection environment, the contribution of radiation is equally signicant, and therefore the design must allow for occurring both convection and radiation. As an application, it can be mentioned to stamped heat sink or extruded heat sink designed for cooling a transistor. Even if forced convection is employed for cooling, radiation is signi cant if the operating temperatures are high as is the case with a nned regenerator [1,2].
Furthermore, to see some very recent investigations on convective-radiative n heat transfer, the interest readers are referred to [45][46][47]. Mosayebidorcheh et. al. have obtained an optimum design point for n geometry so that heat transfer rate reaches to a maximum value in a constant n volume [45]. In [46], authors have applied spectral collocation method for transient thermal analysis of coupled conductive, convective and radiative heat transfer in the moving plate with temperature dependent properties and heat generation. A spectral element method (SEM) has been developed in [47] in order to solve coupled conductive, convective and radiative heat transfer in moving porous ns of trapezoidal, convex parabolic and concave parabolic pro les.
We consider a longitudinal n of arbitrary pro le attached to a primary surface at temperature T b . Let the n length be L and its thicknesses at the base and at the tip be w b and w t , respectively. Further, let w(x) represent the n thickness at any distance x (measured from the base). Both top and bottom faces of the n interact with the surroundings through convection and radiation. The convection process is characterized by the heat transfer coe cient h and the environment temperature T∞. To describe the surface radiation loss, we assume an emissivity ε and an e ective sink temperature Ts. Assuming one-dimensional conduction, constant thermal parameters, and neglecting nto-base and n-to-n radiation interaction, the governing equation for a unit depth of the n is as follows [2]: where h, k, ε, σ, T∞ and Ts denote convective heat transfer coe cient, thermal conductivity of n material, surface emissivity, Stefan-Boltzmann constant, environment temperature for convection and e ective sink temperature for radiation, respectively. Introducing the dimensionless variables To solve Eq. (2), the pro le function w(x) and the boundary conditions must be speci ed. The pro le shapes chosen for the present work are triangular and concave parabolic. The n base temperature in each case is assumed to be constant T b . For the triangular and parabolic ns, the tip heat uxes are obviously zero. Therefore Triangular n: Concave parabolic n: The problem formulated by Eq. (2) with boundary conditions (3) and (4) have been investigated numerically and semi-analytically by many researchers, see [2,[48][49][50][51] and references therein.
In this article, we propose a new intelligent computational approach to obtain solution for the non-linear second-order boundary value problem (2)-(4). First, we transform the governing equation into an equivalent problem whose boundary conditions are [− , ]. In this way, they are convenient to apply reformed version of Chebyshev polynomials of the rst kind. Then we optimize Chebyshev polynomials of the rst kind to construct approximate series solution with unknown weights. Furthermore, it is set up an optimization problem based on unsupervised error as objective function subject to a tolerance as constraint. This optimization problem is minimized by tuning weights via interior point method. This numerical based technique enables us to overcome the nonlinearity in the mentioned boundary value problem and then to obtain accurate solution. Moreover, in some exactly solvable cases, we compare the approximate solution with the exact one. Also, the n e ciency and heat transfer rate with respect are reported.

High order derivatives of basis functions
Chebyshev polynomials [52] are very useful as orthogonal polynomials on the interval [− , ] of the real line. These polynomials have very good properties in the approximation of functions so that appear frequently in several elds of mathematics, physics and engineering.

. Basic properties of Chebyshev polynomials
The Chebyshev polynomials of the rst kind, known as Tn(x) = cos (n arccos x), can be obtained by means of Rodrigue's formula [53] Tn The Chebyshev polynomials of the rst kind can be developed by means of the generating function too, as follows: The rst two Chebyshev polynomials Tn(x) = and Tn(x) = x are known from (5), all other polynomials Tn(x), n ≥ can be obtained by means of the recurrence formula The derivative of Tn(x) with respect to x can be obtained from The following special values and properties of Tn(x) are well established and will be useful: We can determine the orthogonality properties for the Chebyshev polynomials of the rst kind from our knowledge of the orthogonality of the cosine functions, as π, m = n = . . High order derivatives of Chebyshev polynomials can be represented as a sum of derivatives of g(x) and h(x) as: where k n are the binomial coe cients.
with coe cients: where α is the integer oor function of argument α.
It is natural with the help of the Leibniz Formula as well as Rodrigue's formula to de ne the higher order derivatives of Tn(x) as: Without going into great detail, if we apply the result of Theorem 2.1 into the above equation then we develop a very e ective formula as the nal result [54]: with coe cientsÂ i k given by (14).

Nonlinear optimization model
By the change of variable x → (x+ ), the boundary value problems (2)- (3) and (2)-(4) can be rewritten as and respectively. Now, it is convenient to treat them by Chebyshev polynomials of the rst kind. Moreover, the change of function θ → θ + transforms the problems into and such that the boundary conditions become homogenous.

. Reformed version of Chebyshev polynomials
De neTn , n ≥ aŝ then obviously, from (10), we havê Eq. (9) impliesT Therefore, from Eqs. (26)- (27), we conclude that the boundary conditions (22) and (24) hold. Furthermore, the second derivative of the reformed version of Chebyshev polynomials of the rst kind are given byT where the right hand side can be obtained by the formula (16) when k = , .

. Corresponding optimization problem
De ne a approximate series solution of order M as and consider the number of N regularly distributed nodal points in interval [− , ], namely x i , i = , , ..., N, then we de ne the unsupervised errors as the sum of mean squared errors: for triangular n, and for concave parabolic n. It is worth to mention here that Θ M (x) automatically satisfy boundary conditions (22) and (24). Now, de ne the following optimization problems for triangular and concave parabolic ns, respectively, where ε is a given tolerance. In our approach, the interior point method (IPM) is used for tuning of weights of the approximate series solution (30). IPM belongs to a class of algorithms which are used for treating constrained optimization problems. The technique is based on Karmarkar's algorithm which has been developed by Narendra Karmarkar in 1984 for linear programming resolution [55]. Detailed in formation about the algorithm is available in references [56,57]. IPMs have been applied to many optimization problems in engineering and applied science such as multi-area optimal reactive power ow [58] and economic dispatch problem [59]. The fundamental trait of interior point methods are based on self-concordant barrier functions which play important role in encoding the convex set. In contrast to the classical simplex method, search for an optimal solution is made by traversing the interior of the feasible region and solving a sequence of subproblems [60].

Numerical experiments and comparison
In this section, we show the results obtained for some case studies which have been adopted from Refs. [2,[48][49][50][51] using proposed method described in the previous sections.
In these examples, N = , the number of total nodal points covering [− , ], is regularly distributed. Moreover, the number of basis function in approximate series solution in Eq. (30) is M = . The obtained solutions can be compared to those of Refs. [2,[48][49][50][51] and references therein. All approximate solutions reported here obtained in seconds by MATLAB softwares programm, therefore the method is highly robust.
MATLAB provides an e cient optimization toolbox that contains functions for nding minimum of a multivariable function while satisfying constraints. The toolbox includes solvers that perform optimization on the various types of linear or nonlinear problems. The function, fmincon(·), of this toolbox is a general, multipurpose optimizer that well tested and frequently used to solve nonlinear programming problems with general equality, inequality, and bound constraints of small, medium, and large scale. To handle optimization problem (33)-(34), we use fmincon(·) augmented to the interior point method (IPM) as described in the previous section. Figures 1 -3 show temperature distribution versus x for di erent values of Biot number Bi = . , . , . , . , , , and α = , , and Nr = , . , when θs = θ∞ = . in the case of triangular n. The same graphs for the case of concave parabolic n are plotted in Figures 4 -6.       The heat transfer rate q (per unit depth) is given by which is in dimensionless form as Fin e ciency is the ratio of the real heat transfer rate to the ideal heat transfer rate for a n of in nite thermal conductivity which can be rewritten in dimensionless form as We have reported dimensionless heat transfer rate and the n e ciency in the cases of the triangular and concave parabolic n for di erent Biot number, radiationconduction number and α in Tables 1-4

Conclusions
In this article, the problem of the evaluation of heat transfer rate from convecting-radiating n in the cases of triangular and concave parabolic shapes has been investigated.     We have considered one-dimensional, steady conduction in the n and neglected radiative exchange between adjacent ns and its primary surface.
It has been proposed a new intelligent computational technique to obtain approximate solution for the mentioned problem. First, the governing equation is transformed into an equivalent problem whose boundary conditions are homogeneous in interval [− , ]. Then, it is optimized Chebyshev polynomials of the rst kind to construct approximate series solution with unknown weights. Furthermore, by de ning an optimization problem and minimizing it, all weights are obtained via interior point method. As a result, we have reported heat transfer rate and the n e ciency in the cases of the triangular and concave parabolic n for di erent Biot number and radiationconduction number with desired order of accuracy.
The method includes three steps: The rst and most important step is to nd Reformed Version of Chebyshev Polynomials i.e. Eq. (25) so that they satisfy the boundary conditions. The second step is to construct the optimizations problems (33) or (34) and the nal step is to demand fmincon(·) augmented to interior point algorithm using MATLAB. It has been revealed through test studies that the method is highly robust and reliable.

Acknowledgement:
The authors would like to thank two anonymous referees for their valuable comments and helpful suggestions which have improved the quality of the paper. n'th Chebyshev polynomials of the rst kind ϵ i (N, α) Objective function in optimization model Greeks symbols σ Stefan-Boltzmann constant η n e ciency α ratio of length to one-half base thickness ε surface emissivity θ dimensionless temperature θs dimensionless e ective sink temperature for radiation θ∞ dimensionless environment temperature for convection